| L(s) = 1 | − 3·2-s − 3-s + 7·4-s − 5-s + 3·6-s − 3·7-s − 15·8-s + 3·10-s + 9·11-s − 7·12-s − 9·13-s + 9·14-s + 15-s + 30·16-s + 2·17-s − 10·19-s − 7·20-s + 3·21-s − 27·22-s − 4·23-s + 15·24-s + 27·26-s − 21·28-s − 10·29-s − 3·30-s + 8·31-s − 57·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 7/2·4-s − 0.447·5-s + 1.22·6-s − 1.13·7-s − 5.30·8-s + 0.948·10-s + 2.71·11-s − 2.02·12-s − 2.49·13-s + 2.40·14-s + 0.258·15-s + 15/2·16-s + 0.485·17-s − 2.29·19-s − 1.56·20-s + 0.654·21-s − 5.75·22-s − 0.834·23-s + 3.06·24-s + 5.29·26-s − 3.96·28-s − 1.85·29-s − 0.547·30-s + 1.43·31-s − 10.0·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1052488644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1052488644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) | |
| 11 | $C_4$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | |
| good | 2 | $C_2^2:C_4$ | \( 1 + 3 T + p T^{2} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.2.d_c_a_b |
| 5 | $C_2^2:C_4$ | \( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 11 p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.5.b_b_l_bk |
| 7 | $C_4\times C_2$ | \( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 15 p T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.d_c_ap_ach |
| 13 | $C_2^2:C_4$ | \( 1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 115 p T^{5} + 18 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.j_s_ael_abej |
| 17 | $C_2^2:C_4$ | \( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 20 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ac_an_au_nd |
| 19 | $C_2^2:C_4$ | \( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.k_v_acs_asb |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.e_cg_hs_cxv |
| 29 | $C_4\times C_2$ | \( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 200 p T^{5} + 31 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.k_bf_hs_csb |
| 31 | $C_2^2:C_4$ | \( 1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 46 p T^{5} + 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ai_d_abu_btf |
| 37 | $C_2^2:C_4$ | \( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 155 p T^{5} - 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.d_as_fz_ctf |
| 41 | $C_2^2:C_4$ | \( 1 - 23 T + 208 T^{2} - 961 T^{3} + 3975 T^{4} - 961 p T^{5} + 208 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ax_ia_abkz_fwx |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.aq_jy_adie_bbnv |
| 47 | $C_2^2:C_4$ | \( 1 + 3 T - 43 T^{2} - 45 T^{3} + 2116 T^{4} - 45 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.d_abr_abt_ddk |
| 53 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ag_x_eq_abyt |
| 59 | $C_2^2:C_4$ | \( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 530 p T^{5} + 131 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.u_fb_uk_fsd |
| 61 | $C_2^2:C_4$ | \( 1 - 3 T - 7 T^{2} - 441 T^{3} + 4900 T^{4} - 441 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ad_ah_aqz_hgm |
| 67 | $D_{4}$ | \( ( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.ac_cp_ahs_pcj |
| 71 | $C_2^2:C_4$ | \( 1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 819 p T^{5} + 253 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.bb_jt_bfn_dw |
| 73 | $C_2^2:C_4$ | \( 1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 130 p T^{5} - 57 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ag_acf_fa_hbd |
| 79 | $C_2^2:C_4$ | \( 1 - 5 T + 6 T^{2} - 715 T^{3} + 9821 T^{4} - 715 p T^{5} + 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.af_g_abbn_ont |
| 83 | $C_2^2:C_4$ | \( 1 - 21 T + 88 T^{2} + 915 T^{3} - 13199 T^{4} + 915 p T^{5} + 88 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.av_dk_bjf_atnr |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.au_ry_aibg_dvhz |
| 97 | $C_2^2:C_4$ | \( 1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 6655 p T^{5} + 537 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.bh_ur_jvz_ebii |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.42794079269473768134228259268, −12.24047193025691409816799142577, −11.91238724455998699638727578856, −11.72925486511160377615403281474, −11.43972178294170517121552360496, −10.83054363188546449216534337670, −10.67312962199266996085002590042, −10.27991495870979554002009742288, −10.04425102563079848458115774396, −9.311555208116823992380309691437, −9.248471303729656249105420811613, −9.135631565494108772424258271424, −9.020062748017145480790623808267, −7.935894424972931507535866257347, −7.73089604144674073321104060504, −7.60256254186797441362040714584, −6.93042874817741200778147271705, −6.65892509575430838722444846410, −6.14354561328573155680917510391, −6.13043416985269818282051426535, −5.57004126017888279944990882501, −4.17616876775155368877715514137, −4.08107322614519672440380098907, −2.94245368695342572678930183445, −2.21987324717829730221720149871,
2.21987324717829730221720149871, 2.94245368695342572678930183445, 4.08107322614519672440380098907, 4.17616876775155368877715514137, 5.57004126017888279944990882501, 6.13043416985269818282051426535, 6.14354561328573155680917510391, 6.65892509575430838722444846410, 6.93042874817741200778147271705, 7.60256254186797441362040714584, 7.73089604144674073321104060504, 7.935894424972931507535866257347, 9.020062748017145480790623808267, 9.135631565494108772424258271424, 9.248471303729656249105420811613, 9.311555208116823992380309691437, 10.04425102563079848458115774396, 10.27991495870979554002009742288, 10.67312962199266996085002590042, 10.83054363188546449216534337670, 11.43972178294170517121552360496, 11.72925486511160377615403281474, 11.91238724455998699638727578856, 12.24047193025691409816799142577, 12.42794079269473768134228259268
